The Behavior of Thin Walled Structures: Beams, Plates, and Shells

193 Pages·1988·3.526 MB·English

by  Jack R. Vinson (auth.)

The Behavior of Thin Walled Structures: Beams, Plates, and Shells
MECHANICS OF SURFACE STRUCTURES
Editors: W. A. Nash and G. /E. Oravas
I. P. Seide, Small Elastic Deformations of Thin Shells. 1975
ISBN 90-286-0064-7
2. V. Pane, Theories of Elastic Plates. 1975
ISBN 90-286-0J04-X
3. J. L. Nowinski, Theory of Thermoelasticity with Applications. 1978
ISBN 90-286-0457-X
4. S. Lukasiewicz, Local Loads in Plates and Shells. 1979
ISBN 90-286-0047-7
5. V. Flirt, Statistics, Formfinding and Dynamics of Air-Supported Membrane
Structures. 1983
ISBN 90-247-2672-7
6. Yeh Kai-yuan, ed., Progress in Applied Mechanics. The Chien Wei-zang Anniversary
Volume. 1986
ISBN 90-247-3249-2
7. R. Negrutiu, Elastic Analysis of Slab Structures. 1986
ISBN 90-247-3367-7
8. J. R. Vinson, The Behavior of Thin Walled Structures: Beams, Plates, and Shells.
1989
ISBN 90-247-3663-3
The Behavior of
Thin Walled Structures:
Beams, Plates, and Shells
By
Jack R. Vinson
Department of Mechanical Engineering
University of Delaware
Newark, Del., USA
Kluwer Academic Publishers
Dordrecht / Boston / London
Library of Congress Cataloging-in-Publication Data
Vi'nson, Jack R., 1929-
The behavior of thin walled structures.
(Mechanics of surface structures j 8)
Includes bibliographies and ind~x.
1. Thin-walled structures. 1. Title. II. Series:
Mechanics of surface structures ; v. 8.
TA660.TSVS6 1988 624.1'77 87-34773
ISBN-13: 978-94-010-7747-7 e-ISBN-13: 978-94-009-2774-2
001: 10.1007/978-94-009-2774-2
Published by Kluwer Academic Publishers,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates
the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
All Rights Reserved
<D 1989 by Kluwer Academic Publishers
Softcover reprint of the hardcover 1s t edition 1989
No part of the material protected by this copyright notice may be reproduced or utilized
in any form or by any means, electronic or mechanical, including photocopying, recording,
or by any information storage and retrieval system, without written permission from the
copyright owners.
To my beautiful wife, Midge, for her love,
encouragement, patience, and direct assistance
that made this textbook possible.
Contents
Preface Xl
1. Equations of Linear Elasticity in Cartesian Coordinates 1
1.1 Stresses 1
1.2 Displacements 2
1.3 Strains 2
1.4 Isotropy and Its Elastic Constants 3
1.5 Equilibrium Equations 4
1.6 Stress-Strain Relations 5
1.7 Linear Strain-Displacement Relations 6
1.8 Compatibility Equations 7
1.9 Summary 7
1.10 References 8
1.11 Problems 8
2. Derivation of the Governing Equations for Beams and Rectangular
Plates 9
2.1 Assumptions of Plate Theory 9
2.2 Derivation of the Equilibrium Equations for a Plate 10
2.3 Derivation of Plate Moment-Curvature Relations and
Integrated Stress Resultant-Displacement Relatiohs 14
2.4 Derivation of the Governing Equations for a Plate 16
2.5 Boundary Conditions 19
2.6 Stress Distribution within a Plate 22
2.7 References 22
2.8 Problems 23
3. Beams and Rods 24
3.1 General Remarks 24
3.2 Development of the Governing Equations 24
3.3 Solutions for the Beam Equation 27
3.4 Stresses in Beams - Rods - Columns 28
vii
Vlll Contents
3.5 Example: Clamped-Clamped Beam with a Constant Lateral
Load, q(x) = - qo 28
3.6 Example: Cantilevered Beam with a Uniform Lateral Load,
q(x) = -qo 29
3.7 Example: Simply Supported Beam with a Uniform Load over
Part of Its Length 30
3.8 Beam with an Abrupt Change in Stiffness 32
3.9 Beam Subjected to Concentrated Loads 33
3.10 Solutions by Green's Functions 35
3.11 Tapered Beam Solution Using Galerkin's Method 37
3.12 Problems 40
4. Solutions to Problems of Rectangular Plates 41
4.1 Some General Solutions to the Biharmonic Equation 41
4.2 Double Series Solution (Navier Solution) 44
4.3 Single Series Solution (Method of M. Levy) 46
4.4 Example of Plate with Edges Supported by Beams 50
4.5 Summary 52
4.6 References 53
4.7 Problems 53
5. Thermal Stresses in Plates 57
5.1 General Considerations 57
5.2 Derivation of the Governing Equations for a Thermoelastic
Plate 58
5.3 Boundary Conditions 61
5.4 General Treatment of Plate Nonhomogeneous Boundary
Conditions 63
5.5 Thermoelastic Effects on Beams 66
5.6 Self-Equilibration of Thermal Stresses 67
5.7 References 68
5.8 Problems 68
6. Circular Plates 70
6.1 Introduction 70
6.2 Derivation of the Governing Equations 71
6.3 Axially Symmetric Circular Plates 75
6.4 Solutions for Axially Symmetric Circular Plates 75
6.5 Circular Plate, Simply Supported at the Outer Edge, SUbjected
to a Uniform Lateral Loading, Po 77
6.6 Circular Plate, Clamped at the Outer Edge, Subjected to a
Uniform Lateral Loading, Po 78
6.7 Annular Plate, Simply Supported at the Outer Edge, SUbjected
to a Stress Couple, M, at the Inner Boundary 78
6.8 Annular Plate, Simply Supported at the Outer Edge, Subjected
to a Shear Resultant, Qo, at the Inner Boundary 79
Contents ix
6.9 General Remarks 79
6.10 Problems 82
7. Buckling of Columns and Plates 86
7.1 Derivation of the Plate Governing Equations for Buckling 86
7.2 Buckling of Columns Simply Supported at Each End 89
7.3 Column Buckling with Other Boundary Conditions 91
7.4 Buckling of Plates Simply Supported on All Four Edges 92
7.5 Buckling of Plates with Other Loads and Boundary Conditions 97
7.6 References 100
7.7 Problems 101
8. The Vibrations of Beams and Plates 102
8.1 Introduction 102
8.2 Natural Vibrations of Beams 103
8.3 Natural Vibrations of Plates 104
8.4 Forced Vibrations of Beams and Plates 105
8.5 References 105
8.6 Problems 106
9. Energy Methods in Beams, Columns and Plates 107
9.1 Introduction 107
9.2 Theorem of Minimum Potential Energy 107
9.3 Analysis of Beams Subjected to a Lateral Load 108
9.4 The Buckling of Columns 111
9.5 Vibration of Beams 112
9.6 Minimum Potential Energy for Rectangular Plates 113
9.7 The Buckling of a Plate under Uniaxial Load, Simply
Supported on Three Sides, and Free on an Unloaded Edge 114
9.8 Functions to Assume in the Use of Minimum Potential Energy
for Solving Beam, Column, and Plate Problems 117
9.9 Problems 118
10. Cylindrical Shells 121
10.1 Cylindrical Shells under General Loads 121
10.2 Circular Cylindrical Shells under Axially Symmetric Loads 126
10.3 Edge Load Solutions 129
10.4 A General Solution for Cylindrical Shells under Axially
Symmetric Loads 132
10.5 Sample Solutions 134
10.6 Circular Cylindrical Shells under Asymmetric Loads 140
10.7 Shallow Shell Theory (Donnell's Equations) 141
10.8 Inextensional Shell Theory 145
10.9 Membrane Shell Theory 148
10.10 Examples of Membrane Theory 151
10.11 References 152
10.12 Problems 153
x Contents
11. Elastic Stability of Shells 156
11.1 Buckling of Isotropic Circular Cylindrical Shells under Axially
Symmetric Axial Loads 156
11.2 Buckling of Isotropic Circular Cylindrical Shells under Axially
Symmetric Axial Loads and an Internal Pressure 160
11.3 Buckling of Isotropic Circular Cylindrical Shells under Bending 161
11.4 Buckling of Isotropic Circular Cylindrical Shells under Lateral
Pressures 161
11.5 Buckling of Isotropic Circular Cylindrical Shells in Torsion 162
11.6 Buckling of Isotropic Circular Cylindrical Shells under
Combined Axial Loads and Bending Loads 163
11.7 Buckling of Isotropic Circular Cylindrical Shells under
Combined Axial Load and Torsion 163
11.8 Buckling of Isotropic Circular Cylindrical Shells under
Combined Bending and Torsion 164
11.9 Buckling of Isotropic Circular Cylindrical Shells under
Combined Bending and Transverse Shear 164
11.10 Buckling of Isotropic Circular Cylindrical Shells under
Combined Axial Compression, Bending and Torsion 165
11.11 Buckling ofIsotropic Spherical Shells under External Pressure 165
11.12 Buckling of Anisotropic and Sandwich Cylindrical Shells 165
11.13 References 165
11.14 Problems 166
12. The Vibration of Cylindrical Shells 168
12.1 Governing Differential Equations for Natural Vibrations 168
12.2 Hamilton's Principle for Determining the Natural Vibrations of
Cylindrical Shells 169
12.3 Reference 170
Appendix 1. Properties of Useful Engineering Materials 171
Appendix 2. Answers to Selected Problems 173
Index 179
Preface
This book is intended primarily as a teaching text, as well as a reference for individual
study in the behavior of thin walled structural components. Such structures are
widely used in the engineering profession for spacecraft, missiles, aircraft, land-based
vehicles, ground structures, ocean craft, underwater vessels and structures, pressure
vessels, piping, chemical processing equipment, modern housing, etc. It presupposes
that the reader has already completed one basic course in the mechanics or strength
of materials. It can be used for both undergraduate and graduate courses.
Since beams (columns, rods), plates and shells comprise components of so many
of these modern structures, it is necessary for engineers to have a working knowledge
of their behavior when these structures are subjected to static, dynamic (vibration and
shock) and environmental loads.
Since this text is intended for both teaching and self-study, it stresses fundamental
behavior and techniques of solution. It is not an encyclopedia of all research or design
data, but provides the reader the wherewithal to read and study the voluminous
literature.
Chapter 1 introduces the three-dimensional equations oflinear elasticity, deriving
them to the extent necessary to treat the following material. Chapter 2 presents, in
a concise way, the basic assumptions and derives the governing equations for
classical Bernoulli-Euler beams and plates in a manner that is clearly understood.
In Chapter 3, the solutions for beam problems are treated for a variety of commonly
occurring static loads. In this chapter, Green's functions are developed and Galer
kin's method are employed to illustrate these two powerful methods of solution.
In Chapter 4, both the N avier and Levy methods of solution for flat plates are
shown, along with numerous solutions and tabulations of results for problems
frequently encountered.
Because thermal loadings are so often involved in practical structures, a compre
hensive discussion and treatment of thermal stresses and deformations is given in
Chapter 5 for beams and plates, including the complexities of nonhomogeneous
boundary conditions that result. The important fact that in many cases stresses due
to thermal gradients are self equilibrating is illustrated. The analogous moisture
effects associated with structures composed of polymeric materials is also discussed.
Chapter 6 introduces the theory and treats numerous problems associated with
circular plates, because they are encountered in so many structural applications.
Xl